# Calculate the value of this real+imaginary expression:

1. Apr 15, 2013

### 1question

1. The problem statement, all variables and given/known data
Let z = 2 + i and u = -3-3i. Calculate the value of |z/u|.

2. Relevant equations

None.

3. The attempt at a solution

|z/u|=|(2 + i)/(-3-3i)|
= |(2 + i)/(-3-3i)*(-3+3i/(-3+3i)| --- if it is not clear, I'm multiplying by the conjugate.
= |(-6+6i-3i+3i^2)/(9-9i+9i-9i^2)|
= |(3i^2+3i-6)/(-9i^2+9)|
= |(3i-9)/18|
= |-1/2+i/6|
= 1/2+i/6

Did I take care of the absolute value part correctly? Thanks.

2. Apr 15, 2013

### Staff: Mentor

The absolute value is a real number, not a complex number. Apart from the last step, it looks fine, but it can be done in an easier way.

3. Apr 15, 2013

### 1question

OK, so any hint as to how to convert it to a real number? Also, why does it have to be a real number? What does being in absolute terms have to do with being real/imaginary?

4. Apr 15, 2013

### Staff: Mentor

Check how "absolute value" is defined for complex numbers.
Follows from the definition.

5. Apr 15, 2013

### micromass

Staff Emeritus
By the way, it's called a modulus and not an absolute value. Absolute value seems to be reserved for $\mathbb{R}$.

6. Apr 15, 2013

### 1question

Ok.

= |-1/2+i/6|
= sqrt((1/2)^2+(1/6)^2)
= srqt(1/4+1/36)
= sqrt(10)/6

Is that it?

7. Apr 15, 2013

### SammyS

Staff Emeritus
That's correct.

It's also true that |z/u| = |z|/|u| .

$\displaystyle |2+i| = \sqrt{5}$

$\displaystyle |-3-3i| = 3\sqrt{2}$

...

8. Apr 15, 2013

### 1question

Thanks for confirming. That IS faster...

9. Apr 15, 2013

### Fredrik

Staff Emeritus
I'm pretty sure I've seen it called "absolute value". Let me check... yes, Saff & Snider (Fundamentals of complex analysis for mathematics, science, and engineering) begins the definition by saying "The modulus or absolute value of..." and Anton (Linear Algebra) defines the "modulus" and then immediately says "The modulus of z is also called the absolute value of z". Those are the only books I checked. I also think that "absolute value" is the more common term in physics books.

10. Apr 15, 2013

### micromass

Staff Emeritus
I see... I never knew that, I thought modulus was more common. Thanks!